1. What is the Focal Point of a Parabola?

The focal point (or focus) of a parabola is a fixed point that, along with the directrix, serves to define the parabola. All points on the parabola are equidistant to the focus and the directrix.

2. How Does the Calculator Work?

The calculator uses the standard parabola focal point formula:

Where:

• \( x \) — X coordinate on the parabola
• \( y \) — Y coordinate on the parabola

Explanation: For a parabola that opens upwards or downwards with vertex at the origin, the focal length is determined by this relationship between any point (x,y) on the parabola and its focus.

3. Importance of Focal Point Calculation

Details: Knowing the focal point is essential in optics (for parabolic mirrors and antennas), physics (for projectile motion), and engineering (for structural designs).

4. Using the Calculator

Tips: Enter the x and y coordinates of any point on the parabola. The y value must not be zero (as division by zero is undefined).

5. Frequently Asked Questions (FAQ)

Q1: What if my parabola isn't centered at the origin?
A: The formula changes. For a parabola with vertex at (h,k), the formula becomes \( F = (x-h)^2/(4(y-k)) + k \).

Q2: How does this relate to the standard form y = ax² + bx + c?
A: The coefficient 'a' in standard form relates to the focal length f by \( a = 1/(4f) \).

Q3: What's the difference between focus and vertex?
A: The vertex is the "tip" of the parabola, while the focus is an interior point that defines its shape.

Q4: Can this be used for sideways parabolas?
A: No, this formula is for vertical parabolas. For horizontal parabolas, swap x and y in the equation.

Q5: How precise should my inputs be?
A: For most practical purposes, 4 decimal places is sufficient, though the calculator accepts more precise values.